Triple Quantum Schubert Calculus
Abstract
We prove a triple quantum Pieri formula: the product of a quantum double Schubert polynomial S̃u(x;y) by a quantum factorial elementary polynomial Ẽp,k(x;z) in a second set of variables z is a sum over chains in the quantum Bruhat graph, indexed by a quantum analogue of Sottile's relation, whose coefficients are monomials in the quantum parameters times polynomials in the differences yi−zj with nonnegative integer coefficients, and which is Graham-positive at z=y. At y=z=0 it is the quantum Pieri rule of Ciocan-Fontanine and Postnikov; at z=y it is an equivariant quantum Pieri rule for the complete flag variety, indexed by chains in the quantum Bruhat graph rather than, as in the rule of Huang and Li, by the data of a reduction to classical cohomology, and proved directly. A doubled Pieri formula, positive in the same senses, treats multiplication by the quantization of a product of two elementary polynomials in the same variables, which is not the product of the quantizations: it is two iterated Pieri steps with the incompatible pairs of chains omitted. More generally, the triple quantum Schubert structure constants cu,vw,d(y;z), the coefficients of S̃u(x;y)S̃v(x;z) in the basis {S̃w(x;y)}, are polynomials in the yi−zj with nonnegative integer coefficients; the proof of this appears as an appendix. We extend the Peterson-Woodward comparison formula and the quantum-to-classical reduction of Leung-Li and Huang-Li to the triple constants, and we describe and prove correct the algorithms of schubmult, which compute products of arbitrary quantum double Schubert polynomials well beyond the reach of previous methods.
Related
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BibTeX
@misc{samuel2026triple,
title = {{Triple Quantum Schubert Calculus}},
author = {Matthew J. Samuel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31533},
eprint = {2610.31533},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31533},
}
RIS
TY - UNPB
TI - Triple Quantum Schubert Calculus
AU - Matthew J. Samuel
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31533
UR - https://arxiv.org/abs/2610.31533
AB - We prove a triple quantum Pieri formula: the product of a quantum double Schubert polynomial S̃_u(x;y) by a quantum factorial elementary polynomial Ẽ_p,k(x;z) in a second set of variables z is a sum over chains in the quantum Bruhat graph, indexed by a quantum analogue of Sottile's relation, whose coefficients are monomials in the quantum parameters times polynomials in the differences y_i−z_j with nonnegative integer coefficients, and which is Graham-positive at z=y. At y=z=0 it is the quantum Pieri rule of Ciocan-Fontanine and Postnikov; at z=y it is an equivariant quantum Pieri rule for the complete flag variety, indexed by chains in the quantum Bruhat graph rather than, as in the rule of Huang and Li, by the data of a reduction to classical cohomology, and proved directly. A doubled Pieri formula, positive in the same senses, treats multiplication by the quantization of a product of two elementary polynomials in the same variables, which is not the product of the quantizations: it is two iterated Pieri steps with the incompatible pairs of chains omitted. More generally, the triple quantum Schubert structure constants c_u,v^w,d(y;z), the coefficients of S̃_u(x;y)S̃_v(x;z) in the basis {S̃_w(x;y)}, are polynomials in the y_i−z_j with nonnegative integer coefficients; the proof of this appears as an appendix. We extend the Peterson-Woodward comparison formula and the quantum-to-classical reduction of Leung-Li and Huang-Li to the triple constants, and we describe and prove correct the algorithms of schubmult, which compute products of arbitrary quantum double Schubert polynomials well beyond the reach of previous methods.
ER -